‎let $n$ and $k$ be two positive integers‎, ‎$kleq n$ and $a$ be an $n-$square quaternion matrix‎. ‎in this paper‎, ‎some results on the $k-$numerical range of $a$ are investigated‎. ‎moreover‎, ‎the notions of $k$-numerical radius‎, ‎right $k$-spectral radius and $k$-norm of $a$ are introduced‎, ‎and some of their algebraic properties are studied‎.

We propose some refinements for the second inequality in $$ \frac{1}{2}\left\Vert A\right\Vert \le w(A)\le \left\Vert A\right\Vert, where A ∈ B(H). In particular, if is hyponormal, then, by refining Young with Kantorovich constant K K(⋅, ⋅), we show that \frac{1}{2{\operatorname{inf}}_{\left\Vert x\right\Vert =1}\zeta (x)}\left|\left\Vert A\right.\right|++\left|\left.{A}^{\ast}\right\Vert \righ...

We study various inequalities for numerical radius and Berezin number of a bounded linear operator on Hilbert space. It is proved that the pure two-isometry 1 Crawford 0. In particular, we show any scalar-valued non-constant inner function θ, Toeplitz Tθ Hardy space 0, respectively. also shown multiplicative class isometries sub-multiplicative commutants shift. have illustrated these results wi...

They proved several properties and introduced some inequalities. We continue with the study of this generalized numerical radius we develop diverse inequalities involving w_N. also particular cases a fixed N(.), for instance p-Schatten norms. In [A generalization radius. Linear Algebra Appl. 569 (2019)], Abu Omar Kittaneh defined new That is, given norm $N(\cdot)$ on $\bh$, space bounded linear...